Lecture 2: Introduction to the modelling landscape

Irena Papst

Guiding questions

🎯 What is a model?

🎯 What makes a model “good”?

🎯 What is the process of modelling?

🎯 What types of models are there?

🎯 What is a model?

These are all models

\[ y(t) = ke^{rt} \]

\[ Y = \beta_1X + \beta_0 \]

What do all of these things have in common?

All models are representations of reality

What might each of these models represent?

\[ y(t) = ke^{rt} \]

\[ Y = \beta_1X + \beta_0 \]

for instance:

1st eqn could represent:
- bacteria reproducing
- the spread of an infectious disease in the early, initial growth phase (before a significant proportion of the infecteds start recovering)
- the shape of a terrible ramp, where \(t\) is the distance from some fixed starting point and \(y\) is the ramp height
map could represent:
- the city of hamilton, ontario for someone travelling through it
the barbie could represent:
- a specific human body type
- a narrow view of femininity
2nd equation could represent:

-   overall spending $Y$ as a function of personal income $X$

-   my happiness $Y$ as a function of the number of pizza slices eaten $X$ (definitely grows linearly without bound)

🎯 What makes a model “good”?

Model quality depends on model purpose

All models are wrong, but some are useful.

– George E. P. Box (statistician, mid-1900s)

Model quality depends on model purpose

~~All models are wrong, but some are useful.~~

– George E. P. Box (statistician, mid-1900s)

Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.

How much error can you tolerate in your model? Still depends on purpose!

Also, which resources do you have at your disposal?

Example: How long would it take me to walk from my (old) office at McMaster to a lecture hall?

First pass: low resolution

Suppose all I have at my disposal is this campus map:

I know it takes me 10 minutes to walk from my old office to the parking lot N.

Visually estimating the relative distances, I guess it would take me 5 minutes to walk to JHE.

Second pass: higher resolution

Suppose I now have more time and/or resources and I’m able to get (most of) the path I’d usually take measured:

distance = 350 m

average walk speed of an adult = 5 km/h or about 83 m/min

time = distance / speed

estimated time from HH to JHE: 4.2 min or 4 min, 12 seconds

Second pass: higher resolution

Suppose I now have more time and/or resources and I’m able to get (most of) the path I’d usually take measured:

Note: just because we can get a level of precision of 4 mins and 12 seconds out of our model, doesn’t mean we should confuse this for accuracy! There are several uncertainties that can affect our estimate… Can you name some?

Second pass: higher resolution

Suppose I now have more time and/or resources and I’m able to get (most of) the path I’d usually take measured:

May not know the size of errors induced by assumptions under uncertainty, but can at last think about potential effects on estimates (e.g. increase or decrease walking time).

🎯 What is the process of modelling?

Modeling is mapping between the real world and a mathematical and/or statistical framework

It’s an iterative process, and the final mapping depends on the acceptable level of error.

🎯 What types of models are there?

\[ y(t) = ke^{rt} \]

\[ Y = \beta_1X + \beta_0 \]

How could we split these models into two different groups?

mechanistic (rule-based): \[ y(t) = ke^{rt} \]

phenomenological (descriptive):

\[ Y = \beta_1X + \beta_0 \]

Phenomenological models

\[ Y = \beta_1X + \beta_0 \]

describe the state of a system at one or various points in time
trying to capture what the system looks like, not trying to explain why it looks like that

Phenomenological models

\[ Y = \beta_1X + \beta_0 \]

don’t necessarily need any domain-specific knowledge or context to formulate
can use phenomenological models for prediction (e.g. extrapolation), but without any context of how the data are generated, we could be badly wrong about the observed relationship in the region we seek to predict

Mechanistic models

\[ y(t) = ke^{rt} \]

try to come up with rules that explain the observed behaviour of a system
capture how the system behaves (why it looks the way it does at various points in time)
need domain-specific knowledge and context

10.1016/j.cois.2016.07.006

Phenomenological 🤝 Mechanistic

gather descriptions (work with phenomenological models)

propose a mechanism

generate predictions

compare to original descriptions & gather new ones

(if predictions =/= descriptions) propose a new mechanism

repeat 3-5 as needed

Phenomenological 🤝 Mechanistic

10.1371/journal.pclm.0000226

phenomenological and mechanistic models can work together explicitly in the form of hybrid models
- imagine we’re trying to come up with an ecological niche model, where we’re trying to predict the geographic occurrence of species over time
- on one side, we have a correlative model (phenomenological), where we learn about correlations between climate indicators (like location temperature and precipation) and species occurrence in a sampled location and then extrapolate that to other settings (based on the climate metrics) to predict species occurence probabilistically
- on the other side, we have mechanistic models that take climate data paired with biological information (genotype, phenotype, etc) to calculate species fitness in different settings
- in between we have hybrid models, where we take biological information to mechanistically predict variables used for correlation with occurrence that can then be extrapolated to other settings for prediction
mechanistic and phenomenological is just one way to characterize models

Other model characterizations

Dynamic or static
- Is there a time component? dynamic
- Otherwise, static
Discrete or continuous
- in state: categorical or numerical variables
- in time: \(y_{t+1} = c y_t\) vs \(dy/dt = cy\)
- Simulations are discrete (discretized) in time!
Deterministic or stochastic
- Is there noise incorporated in the model?

Summary

🎯 What is a model?

A representation of reality

🎯 What makes a model “good”?

Depends on the purpose of the model

Summary

🎯 What is the process of modelling?

Summary

🎯 What types of models are there?

phenomenological or mechanistic
dynamic or static
discrete or continuous
deterministic or stochastic